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On exponential splines
JOURNAL
ok APPROXIMATION
47: 122-13
THEORY
1 (1986)
On Exponential MAXABU
SAKAI
Splines
AND
A.
RIAZ
USMANI
Department of Mathematics, Faculty of Science, Kagoshima University, Kagoshima 890, Japan C’ommunirated by R. Rojanic Rcccivcd
August
6, 1984; revised
1. IxiTRoDucrrox
AXD
December
DESCRIPTION
24, 1984
OF B-SPI,IX;ES
Among the various classesof splines, the polynomial spline has been received the greatest attention primarly because it admits a basis of Bsplines which can be accurately and efficiently computed. Recently it has been shown that trigonometric and hyperbolic splines also admit B-spline bases([I 1, 21). The object of the present paper is to give B-spline bases for hyperbolic and trigonometric splines which are little different from the hyperbolic and trigonometric ones in [ 1, 21. Throughout this paper, we assumethat m is a natural number and 2 is a positive parameter. First we consider a sequence {d,,j} defined by dm,i= dm - 1,,i+ dm
1,j
1,
dm,,
dm,j = 0
(j<
-1 and jam+
4, = 1,
d2,,
= 2
=
4x,,
=
(m33)
1
1)
cash L,
(1.1)
d,,, = 1.
By a simple calculation, we have (1.2)
.
Now, by making use of the constant spline Q, + ,,i of degree m: for m odd: Q,?
,&c)
=
c j-0
( -
qx
1 K,
-- j)
d,,,],
, J
,
...
122 0021-9045186
S3.00
Copyright c> 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.
we may define a hyperbolic
l/i”)
Csinh
(i(x-(h--2)!--
3x
j),}“?
-A
B-
,
*
1 ;
(1.3)
OK EXPONENTIAL
123
SPLINES
for m even: m+l
Qm+l,ltX)=
1
tl/nmn)
6Wm+l,j
j-0
cash j(x - j) +
[
..I - (i(x -j) + p-2 (m-2)! I.
-l-
Here we shall prove that the hyperbolic B-spline Q,, r,;. is characterized by a convolution process of an exponential function di and a characteristic function x on [O, l), where
THEOREM
qSi(x) = e’”
(O
1)
and
0
(otherwise)
x(x) = 1
(O,
1)
and
0
(otherwise).
(1.3
1.
Qm+ l,ltx) = t-4?.
*i
(1.6)
L)tx)
m-i where * means the convolution tf*g)tx) Proof:
By the definition
of two functions, = jm f(f) .m
i.e.,
gtx - t) dt.
of dm,j, we have (1.7)
SinceQ, +l,A(0) = 0, from above we obtain Qm+l,~tX)=j~~e
1 Qm,j.tt)dt= jc: X(t)
Qm,,tx- tf dt (1.8)
=
(x*Qm,i)tx) = . . . = (~Q2,j.)(x). m-1
By a simple calculation,
we have
AQ*Jx)
= sinh ix,
- cash i sinh i,(x - 1) +
+ sinh I-(x - 2) + sinh Ix = { sinh 42 - x) = 44, *4 d(x). This completes the proof of Theorem 1.
(OdxG
1)
(1
124
SAIL41
AND
USMANI
Next we shall give a trigonometric B-spline &+ 1,1 by replacing the parameter ;1 in the definition of the hyperbolic B-spline Qm + 1,1 by i,l (i = Q): for m odd: m+l &
m+I,i(~)=(-1)(1~2)(m-1)
1
(-l)Gm+,.Jl/Am) [
j=O
-jqx-j)+-
sinA(x-j),
w-“9,
. . . -(-1)(1/m+‘)
F2
(m-2)!
1 ;
(1.10)
for m even: m+l Q
,+l,n(x)=(-1)(“2)”
1
(-l)‘z~+rJ(l/nm)
cosqx-j), C
j=O
_ 1 _ . . . _ (_ p)fm+2)
{4x-A+ Y-* (m-2)! I (1.11)
where {zm,j> is delined by the same recursion formula (1.1) with 2 cos L in the definition of &*. Similarly as in the hyperbolic B-spline, we have: THEOREM
2. 0 m + 1,1(x)
=
(X*X*
. . .
(1.12)
*x*Qu *$-ii.)(x),
m-l where (#u *d-u)(x)
= sin Ax/L
(06x<
= sin A(2 - x)/A
(1
1)
(1.13)
By making use of Theorems 1 and 2, we may easily have the properties of these hyperbolic and trigonometric B-splines similar to those of the polynomial ones ([3]). In addition, we have the following theorems that imply 1, x ,..., Y-‘, cash Ax and sinh ~x~Span(Q,+,,~(x-j)),? --co. THEOREM
3. For m 2 2, 1, x ,..., x”-‘~Span(Q,+,,,(x-j)}i”=
--OO.
(1.14)
ON
ProoJ:
By the definition
EXPONEKTIAI,
of the characteristic function, we have f ix
A successive convolution yields
x(x-j)=
(1.15)
1.
--cr:
of this partition
of unity and I,..., x, 4j. and d, -j.
f Q,+,,;(X-i)=(l*~~j.*~ j- --u2
j.)(x) = {sinh(& L)/(f ,I)}’
In addition,
125
SPLINES
(m 3 2).
(1.16)
since j - (j - 1) = 1, from above we have {sinh(i i-)/(4 I-)}’
= Integration
f
j-
a:
(1.17)
jQin+2,j.(X-j).
of the above equation from 0 to x gives i
f
(x-j)-
jQ,+,,j.
f j;
:x:
jQnz+2.A-.d --a:
= (sinh(+ ;2)/(i ;*))}‘x
(1.18)
where
a: c Qm+2.i.(-A ,=. o(j = -(Q,+*,i.(l)+2Qm+*,i.(2)+ = -($ m + l){sinh(i During the above computation,
... +fm+l)Qm+2,~("+1))
L)/($ I.))“. we used
Qm t 2,2(j) = Qm+2,j.(m + 2 --.i) fQ j=
--a:
(1.19)
,~), 2,1(x - j) = (sinh(i i”)/( 41”) >*.
(1.20)
126
SAKAIANDUSMANI
Thus we have for m 2 2 f j=
-a
(j+$m+l)Q
,+,,,(x-A=
{sinh(i~Y(5~))2x
(1.21)
i.e.,
x~Span{Q,+l,,(x-j)}i”=
--oD (for ma3).
(1.22)
By making use of a simple identity: k-2
k-l
kpJIo(j+p)=
II p=o
k-l (j+d-
l-I
(1.23)
(j+p-1)
p=o
i.e., 2j= (j+ 1) j- j(jW+
inductively THEOREM
1) j= (j+2)(j+ . .. 2
1) j-
1) (j-t 1) j(j-
1)
we have the desired result. 4. For m 2 2, f j=
Q,,n(x-j)coshA(j++m) -02 =[{(2/A)sinh~A.}“-3cosh$1]cosh1x (1.24)
j=
2 Q,,,(x-j)sinhA(j++m) -02 = [{(2/,I)sinh+,I}“-3coshtA]
sinhAx.
ProoJ: For m = 2, by an elementary computation we have the above relations. Multiplying the first relation by 2 sinh &1, we have for m 3 3 j=f?,
Qk+1,~(x-j)sinhI{j+f(m+l)) ={(2/A)sinh+1}“-32sinh$ilcos~AcoshIlx
where
Q:,.,,, (x) = Q,,,(x) - Qm,a(x- 1).
(1.25)
ON
Integration
EXPONElrjTIAL
127
SPLIKNES
of the above equation from 0 to x yields m+ I,j.(X-j) sinh A(j+ $(m + 1)) + c 30 = { (2/A) sinh 1 A>M *cash 4 L sinh ix.
fQ j-
(1.26)
Here we shall prove the above constant c to be zero. By Theorem 1, we obtain Q mt
,,j.(X)=d[
(1.27)
2Qm--,(t)~(x--)~~
where
i.e.,Q, 1 is
the polynomial B-spline of degree m - 2, and I&X) = (dj. *d j.)(X). From (1.27), by a simple calculation we get (L’*
Operating have
-
A*)
Q m + 1,2(x)
=
Qm- I,,~(x)- 2 cashi Qm 1,2.(X - 1) (1.28) (m33). + Qm-.I,L(X- 21
the differential operator (D* - A2) to the both sides of f 1.26), we
j;;
z [sinhA{(j+$(m+ -* -2coshAsinhi,((j--1
++(m+f)}
+sinhA((j-2+i(m+l))]
XQWi ,,Jx - j) - c?“* = 0.
(1.29)
Since the coefficient of Qnl ,,I(x - j) 1s . 1‘d en ti ca11y zero, we have the desired second relation with m replaced by m + 1. Similarly we have the first relation with m + 1 from the second with m. For 0, + l,j.(X), we have z j=
Q,,
Jx-j)
= (sin(+ R)/(t ;1)}”
(m Z 2)
(1.30)
cc
from which follows 1 ESPan{D,+~.(x-j)jjZ for
i#2kn
-cm (k = I, 2,...).
(m22) (1.31)
SAKAIAND USMANI
128 In addition,
as in the proof of Theorem 3 we have x,x2 ,..., x”-2~Span{&,+,,,(x-j)},E for II # 2kn
For Q,,,(x)
--co
(k = 1, 2,...).
(1.32)
(m > 2), we have
j=lfm
L,n(x-~m~(~+~~)
= [{(2/1)sin+jl}
M-3COS + A] cos Ax (1.33)
f j=
Q,,n(x-j)sinA(j+$m)
--m = [{(2/il)sin+/2}
mp3cos 4 A] sin Lx
from which follows cos Ix and sin Ix E Span(&,(x forI#krc
-j)}~=
--m
(m32)
(k = 1, 2,...).
(1.34)
2. AN APPLICATION OF A HYPERBOLICSPLINEOF DEGREE4 TO A NUMERICALSOLUTIONOF A SIMPLEPERTURBATIONPROBLEM We are concerned with numerical bation problem:
solution
of a simple singular pertur-
(OGXG 1) &YUb)- Y(X) = g(x) Y(O)= a, vc1,=a with O
(l
y,=P
with
A= h/A
where for a natural number IZ, h = l/n and gj= g(jh).
(2.1) (2.2)
in E, of the
(2.3) (2.4)
ON
EXPOM3JTIAL
SPLINES
Now, by making use of the hyperbolic sider a spline function of the form
129
B-spline es,), of degree 4 we con-
S(X) =j=Bxl-4 xjQs,j.(xlh -j),
i. = h/&
(2.5)
with undetermined coefficients (Lo, 2. 3,..., x11 1). The above s(x) will be an approximate solution if it satisfies &S;
-
.yi =
gj
sg = cl,
(l
(2.6)
s, = 8.
(2.7)
In order to transform the above collocation method (2.6)~-(2.7) to a difference method, we shall require the following consistency relation obtained by use of the similar technique for the polynomial spline: (#I
h
2(a2,4sj
1+a2,3sj+a2,2sj;l+s2,,~~j-2)
=
(a 0.4.q
~$n,,,3.s;'+a,,zsl'+1+a,,,~s~+,)
(2.8)
where u,,~ = Qik](j) (1
(2.9)
where p(i) =
1*(cosh iv - 1) > 12. cash i. -. I - 3 A”
By means of this short term consistency relation, (2.6)-(2.7) is equivalent to a difference method: Ep(Sj+l-2Sj+LS,j-l) hZ
sg = SC,
sn=B
the collocation
-{Sj+l+(,L-2)sj+Sj
=gj-1+(Pp22)gj+Kj-
1
with
(2.10) method
1) (1 <,j
p = p(h/J).
1)
(2.1%) (2.12)
130
SAKAI
AND
TABLE
USMANI
(E = 10m4)
Observed Errors at Mesh Points in Collocation Method (2.11)-(2.12) and Miller’s Difference Method (2.3)-(2.4) Method x\h 0.05 0.1 0.2 0.3 0.4 0.5
Collocation
Difference
l/20
l/40
l/20
l/40
0.922-5* 0.790-5 0.302-5 - 0.3.2-5 - 0.790-5 -0.977-5
0.955-6 0.818-6 0.313-6 -0.313-6 -0.818-6 -0.101-5
-0,301-l -0.132-2 -0.310-5 -0.498-5 -0.130-4 -0.161-4
-0.840-2 -0.181-3 -0.120-5 -0.125-5 -0.361-5 - 0.404-5
* We denote 0.922 x 10m5 by 0.922-5, and the errors mean (exact values) - (approximate values).
The solution of (2.1)-(2.2) would be dominated by terms of e*X’d, and so in order to derive a lumped mass system of the above difference scheme we let (sj+l+gj+l)+(sj-,+gj-,)
g (eh/d + e-‘/A)
x
(~j + gj) = 2 cosh(h/&)(Jj+
gj).
(2.13)
Since p(h/&)
- 2 + 2 cosh(h&)
= { sin;y ‘)}*
p(h/,,&),
(2.14)
we have a lumped mass system of (2.6)-(2.7) which is identical with the above Miller’s difference scheme (2.3)-(2.4). Now we consider an application of the difference scheme (2.1 l )-(2.12) to a simple two point boundary value problem: &yN- y = cos27Lx+ 2Z2E cos 271x,
y(0) = y( 1) = 0.
The exact solution is given by
y(x) =exP((x- 1)/J? + ew(--xl&) 1 + exp( - l/d)
-cos2nx.
(2.15)
O;\;
EXPONENTIAL
SPLINES
13i
REFERENCES
1. T. LYCHE A?iD R. WINTHE~, A Approx.
Theory
25 (1979),
stable
recurrence relation for trigonometric
B-sphnes, 9.
266.-279.
2. L. L. SCHUMAKER,On hyperbolic splines, J. Approx. Theory 38 (1983) 144-166. 3. L. L. SCHUMAKER,“Spline Functions: Basic Theory,” Wiley, New York, 1981. 4. J. J. MILLER, On the convergence, uniformly in E. of difference schemes for a two point boundary singular perturbation problem, in “Proceedings, Conf., Nijmegen,” Academic Press, New York, 1979. 5. J. STO~R AKD R. BULIRSCH,“Introduction to Numerical Analysis,” Springer-Verlag, Berlin,/ New Y.ork/Heidelberg, 1979.
ok APPROXIMATION
47: 122-13
THEORY
1 (1986)
On Exponential MAXABU
SAKAI
Splines
AND
A.
RIAZ
USMANI
Department of Mathematics, Faculty of Science, Kagoshima University, Kagoshima 890, Japan C’ommunirated by R. Rojanic Rcccivcd
August
6, 1984; revised
1. IxiTRoDucrrox
AXD
December
DESCRIPTION
24, 1984
OF B-SPI,IX;ES
Among the various classesof splines, the polynomial spline has been received the greatest attention primarly because it admits a basis of Bsplines which can be accurately and efficiently computed. Recently it has been shown that trigonometric and hyperbolic splines also admit B-spline bases([I 1, 21). The object of the present paper is to give B-spline bases for hyperbolic and trigonometric splines which are little different from the hyperbolic and trigonometric ones in [ 1, 21. Throughout this paper, we assumethat m is a natural number and 2 is a positive parameter. First we consider a sequence {d,,j} defined by dm,i= dm - 1,,i+ dm
1,j
1,
dm,,
dm,j = 0
(j<
-1 and jam+
4, = 1,
d2,,
= 2
=
4x,,
=
(m33)
1
1)
cash L,
(1.1)
d,,, = 1.
By a simple calculation, we have (1.2)
.
Now, by making use of the constant spline Q, + ,,i of degree m: for m odd: Q,?
,&c)
=
c j-0
( -
qx
1 K,
-- j)
d,,,],
, J
,
...
122 0021-9045186
S3.00
Copyright c> 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.
we may define a hyperbolic
l/i”)
Csinh
(i(x-(h--2)!--
3x
j),}“?
-A
B-
,
*
1 ;
(1.3)
OK EXPONENTIAL
123
SPLINES
for m even: m+l
Qm+l,ltX)=
1
tl/nmn)
6Wm+l,j
j-0
cash j(x - j) +
[
..I - (i(x -j) + p-2 (m-2)! I.
-l-
Here we shall prove that the hyperbolic B-spline Q,, r,;. is characterized by a convolution process of an exponential function di and a characteristic function x on [O, l), where
THEOREM
qSi(x) = e’”
(O
1)
and
0
(otherwise)
x(x) = 1
(O,
1)
and
0
(otherwise).
(1.3
1.
Qm+ l,ltx) = t-4?.
*i
(1.6)
L)tx)
m-i where * means the convolution tf*g)tx) Proof:
By the definition
of two functions, = jm f(f) .m
i.e.,
gtx - t) dt.
of dm,j, we have (1.7)
SinceQ, +l,A(0) = 0, from above we obtain Qm+l,~tX)=j~~e
1 Qm,j.tt)dt= jc: X(t)
Qm,,tx- tf dt (1.8)
=
(x*Qm,i)tx) = . . . = (~Q2,j.)(x). m-1
By a simple calculation,
we have
AQ*Jx)
= sinh ix,
- cash i sinh i,(x - 1) +
+ sinh I-(x - 2) + sinh Ix = { sinh 42 - x) = 44, *4 d(x). This completes the proof of Theorem 1.
(OdxG
1)
(1
124
SAIL41
AND
USMANI
Next we shall give a trigonometric B-spline &+ 1,1 by replacing the parameter ;1 in the definition of the hyperbolic B-spline Qm + 1,1 by i,l (i = Q): for m odd: m+l &
m+I,i(~)=(-1)(1~2)(m-1)
1
(-l)Gm+,.Jl/Am) [
j=O
-jqx-j)+-
sinA(x-j),
w-“9,
. . . -(-1)(1/m+‘)
F2
(m-2)!
1 ;
(1.10)
for m even: m+l Q
,+l,n(x)=(-1)(“2)”
1
(-l)‘z~+rJ(l/nm)
cosqx-j), C
j=O
_ 1 _ . . . _ (_ p)fm+2)
{4x-A+ Y-* (m-2)! I (1.11)
where {zm,j> is delined by the same recursion formula (1.1) with 2 cos L in the definition of &*. Similarly as in the hyperbolic B-spline, we have: THEOREM
2. 0 m + 1,1(x)
=
(X*X*
. . .
(1.12)
*x*Qu *$-ii.)(x),
m-l where (#u *d-u)(x)
= sin Ax/L
(06x<
= sin A(2 - x)/A
(1
1)
(1.13)
By making use of Theorems 1 and 2, we may easily have the properties of these hyperbolic and trigonometric B-splines similar to those of the polynomial ones ([3]). In addition, we have the following theorems that imply 1, x ,..., Y-‘, cash Ax and sinh ~x~Span(Q,+,,~(x-j)),? --co. THEOREM
3. For m 2 2, 1, x ,..., x”-‘~Span(Q,+,,,(x-j)}i”=
--OO.
(1.14)
ON
ProoJ:
By the definition
EXPONEKTIAI,
of the characteristic function, we have f ix
A successive convolution yields
x(x-j)=
(1.15)
1.
--cr:
of this partition
of unity and I,..., x, 4j. and d, -j.
f Q,+,,;(X-i)=(l*~~j.*~ j- --u2
j.)(x) = {sinh(& L)/(f ,I)}’
In addition,
125
SPLINES
(m 3 2).
(1.16)
since j - (j - 1) = 1, from above we have {sinh(i i-)/(4 I-)}’
= Integration
f
j-
a:
(1.17)
jQin+2,j.(X-j).
of the above equation from 0 to x gives i
f
(x-j)-
jQ,+,,j.
f j;
:x:
jQnz+2.A-.d --a:
= (sinh(+ ;2)/(i ;*))}‘x
(1.18)
where
a: c Qm+2.i.(-A ,=. o(j = -(Q,+*,i.(l)+2Qm+*,i.(2)+ = -($ m + l){sinh(i During the above computation,
... +fm+l)Qm+2,~("+1))
L)/($ I.))“. we used
Qm t 2,2(j) = Qm+2,j.(m + 2 --.i) fQ j=
--a:
(1.19)
,~), 2,1(x - j) = (sinh(i i”)/( 41”) >*.
(1.20)
126
SAKAIANDUSMANI
Thus we have for m 2 2 f j=
-a
(j+$m+l)Q
,+,,,(x-A=
{sinh(i~Y(5~))2x
(1.21)
i.e.,
x~Span{Q,+l,,(x-j)}i”=
--oD (for ma3).
(1.22)
By making use of a simple identity: k-2
k-l
kpJIo(j+p)=
II p=o
k-l (j+d-
l-I
(1.23)
(j+p-1)
p=o
i.e., 2j= (j+ 1) j- j(jW+
inductively THEOREM
1) j= (j+2)(j+ . .. 2
1) j-
1) (j-t 1) j(j-
1)
we have the desired result. 4. For m 2 2, f j=
Q,,n(x-j)coshA(j++m) -02 =[{(2/A)sinh~A.}“-3cosh$1]cosh1x (1.24)
j=
2 Q,,,(x-j)sinhA(j++m) -02 = [{(2/,I)sinh+,I}“-3coshtA]
sinhAx.
ProoJ: For m = 2, by an elementary computation we have the above relations. Multiplying the first relation by 2 sinh &1, we have for m 3 3 j=f?,
Qk+1,~(x-j)sinhI{j+f(m+l)) ={(2/A)sinh+1}“-32sinh$ilcos~AcoshIlx
where
Q:,.,,, (x) = Q,,,(x) - Qm,a(x- 1).
(1.25)
ON
Integration
EXPONElrjTIAL
127
SPLIKNES
of the above equation from 0 to x yields m+ I,j.(X-j) sinh A(j+ $(m + 1)) + c 30 = { (2/A) sinh 1 A>M *cash 4 L sinh ix.
fQ j-
(1.26)
Here we shall prove the above constant c to be zero. By Theorem 1, we obtain Q mt
,,j.(X)=d[
(1.27)
2Qm--,(t)~(x--)~~
where
i.e.,Q, 1 is
the polynomial B-spline of degree m - 2, and I&X) = (dj. *d j.)(X). From (1.27), by a simple calculation we get (L’*
Operating have
-
A*)
Q m + 1,2(x)
=
Qm- I,,~(x)- 2 cashi Qm 1,2.(X - 1) (1.28) (m33). + Qm-.I,L(X- 21
the differential operator (D* - A2) to the both sides of f 1.26), we
j;;
z [sinhA{(j+$(m+ -* -2coshAsinhi,((j--1
++(m+f)}
+sinhA((j-2+i(m+l))]
XQWi ,,Jx - j) - c?“* = 0.
(1.29)
Since the coefficient of Qnl ,,I(x - j) 1s . 1‘d en ti ca11y zero, we have the desired second relation with m replaced by m + 1. Similarly we have the first relation with m + 1 from the second with m. For 0, + l,j.(X), we have z j=
Q,,
Jx-j)
= (sin(+ R)/(t ;1)}”
(m Z 2)
(1.30)
cc
from which follows 1 ESPan{D,+~.(x-j)jjZ for
i#2kn
-cm (k = I, 2,...).
(m22) (1.31)
SAKAIAND USMANI
128 In addition,
as in the proof of Theorem 3 we have x,x2 ,..., x”-2~Span{&,+,,,(x-j)},E for II # 2kn
For Q,,,(x)
--co
(k = 1, 2,...).
(1.32)
(m > 2), we have
j=lfm
L,n(x-~m~(~+~~)
= [{(2/1)sin+jl}
M-3COS + A] cos Ax (1.33)
f j=
Q,,n(x-j)sinA(j+$m)
--m = [{(2/il)sin+/2}
mp3cos 4 A] sin Lx
from which follows cos Ix and sin Ix E Span(&,(x forI#krc
-j)}~=
--m
(m32)
(k = 1, 2,...).
(1.34)
2. AN APPLICATION OF A HYPERBOLICSPLINEOF DEGREE4 TO A NUMERICALSOLUTIONOF A SIMPLEPERTURBATIONPROBLEM We are concerned with numerical bation problem:
solution
of a simple singular pertur-
(OGXG 1) &YUb)- Y(X) = g(x) Y(O)= a, vc1,=a with O
(l
y,=P
with
A= h/A
where for a natural number IZ, h = l/n and gj= g(jh).
(2.1) (2.2)
in E, of the
(2.3) (2.4)
ON
EXPOM3JTIAL
SPLINES
Now, by making use of the hyperbolic sider a spline function of the form
129
B-spline es,), of degree 4 we con-
S(X) =j=Bxl-4 xjQs,j.(xlh -j),
i. = h/&
(2.5)
with undetermined coefficients (Lo, 2. 3,..., x11 1). The above s(x) will be an approximate solution if it satisfies &S;
-
.yi =
gj
sg = cl,
(l
(2.6)
s, = 8.
(2.7)
In order to transform the above collocation method (2.6)~-(2.7) to a difference method, we shall require the following consistency relation obtained by use of the similar technique for the polynomial spline: (#I
h
2(a2,4sj
1+a2,3sj+a2,2sj;l+s2,,~~j-2)
=
(a 0.4.q
~$n,,,3.s;'+a,,zsl'+1+a,,,~s~+,)
(2.8)
where u,,~ = Qik](j) (1
(2.9)
where p(i) =
1*(cosh iv - 1) > 12. cash i. -. I - 3 A”
By means of this short term consistency relation, (2.6)-(2.7) is equivalent to a difference method: Ep(Sj+l-2Sj+LS,j-l) hZ
sg = SC,
sn=B
the collocation
-{Sj+l+(,L-2)sj+Sj
=gj-1+(Pp22)gj+Kj-
1
with
(2.10) method
1) (1 <,j
p = p(h/J).
1)
(2.1%) (2.12)
130
SAKAI
AND
TABLE
USMANI
(E = 10m4)
Observed Errors at Mesh Points in Collocation Method (2.11)-(2.12) and Miller’s Difference Method (2.3)-(2.4) Method x\h 0.05 0.1 0.2 0.3 0.4 0.5
Collocation
Difference
l/20
l/40
l/20
l/40
0.922-5* 0.790-5 0.302-5 - 0.3.2-5 - 0.790-5 -0.977-5
0.955-6 0.818-6 0.313-6 -0.313-6 -0.818-6 -0.101-5
-0,301-l -0.132-2 -0.310-5 -0.498-5 -0.130-4 -0.161-4
-0.840-2 -0.181-3 -0.120-5 -0.125-5 -0.361-5 - 0.404-5
* We denote 0.922 x 10m5 by 0.922-5, and the errors mean (exact values) - (approximate values).
The solution of (2.1)-(2.2) would be dominated by terms of e*X’d, and so in order to derive a lumped mass system of the above difference scheme we let (sj+l+gj+l)+(sj-,+gj-,)
g (eh/d + e-‘/A)
x
(~j + gj) = 2 cosh(h/&)(Jj+
gj).
(2.13)
Since p(h/&)
- 2 + 2 cosh(h&)
= { sin;y ‘)}*
p(h/,,&),
(2.14)
we have a lumped mass system of (2.6)-(2.7) which is identical with the above Miller’s difference scheme (2.3)-(2.4). Now we consider an application of the difference scheme (2.1 l )-(2.12) to a simple two point boundary value problem: &yN- y = cos27Lx+ 2Z2E cos 271x,
y(0) = y( 1) = 0.
The exact solution is given by
y(x) =exP((x- 1)/J? + ew(--xl&) 1 + exp( - l/d)
-cos2nx.
(2.15)
O;\;
EXPONENTIAL
SPLINES
13i
REFERENCES
1. T. LYCHE A?iD R. WINTHE~, A Approx.
Theory
25 (1979),
stable
recurrence relation for trigonometric
B-sphnes, 9.
266.-279.
2. L. L. SCHUMAKER,On hyperbolic splines, J. Approx. Theory 38 (1983) 144-166. 3. L. L. SCHUMAKER,“Spline Functions: Basic Theory,” Wiley, New York, 1981. 4. J. J. MILLER, On the convergence, uniformly in E. of difference schemes for a two point boundary singular perturbation problem, in “Proceedings, Conf., Nijmegen,” Academic Press, New York, 1979. 5. J. STO~R AKD R. BULIRSCH,“Introduction to Numerical Analysis,” Springer-Verlag, Berlin,/ New Y.ork/Heidelberg, 1979.